Artificial lift systems, such as sucker rod systems, also referred to as beam pumped systems, are widely used to mitigate the pressure decline in hydrocarbon producing wells over time. Such systems are used in both conventional oil and gas fields and unconventional oil and gas fields (e.g., shale oil and coalbed methane, where formations must be dewatered prior to hydrocarbon production). Artificial lift is installed in wells that are no longer capable of lifting fluids to the surface using the reservoir's natural energy. In these completions, a rod string is connected to a plunger that actuates a ball and seat valve downhole. These wells were originally allowed to run twenty-four hours per day. Eventually operators discovered that this was contributing to the failure of pump components as the pump was often not properly primed with reservoir fluids downhole. For some portion of the day, the reservoir was so depleted that the pump plunger would pound against a low fluid level.
Artificial lift wells that are operated by cyclic pump off controllers, or simply “pump off controllers” or POCs, are stopped and started intermittently based on the controller logic. POCs were invented to offer a way for operators to produce a well for a period of time and stop the pump from reciprocating for the remainder of the day. Type II POCs generally utilize sensors for monitoring pump conditions. For instance, load cells can be used to measure the load on a beam pump, which is an indication of the fluid level in the well. When the fluid level reaches a minimum acceptable limit, the load on the beam pump will be at a maximum acceptable limit. When this limit is reached, a condition referred to as “pump-off,” the POC shuts down production of the well for a period of time to allow fluid to enter the well. This period of time is generally predetermined by the POC operator. As a result, the transient pressure and rate behavior of beam pumped wells is quite complicated.
In the field of reservoir and production engineering, it is frequently important to assess the productivity of an intermittently produced beam pumped hydrocarbon producing well, also referred to herein interchangeably as a well or its associated reservoir, in what is referred to as a well transient test. This test may be conducted to determine reservoir properties including permeability-thickness (kh) and skin (s). Permeability-thickness is the flow capacity of the gross reservoir rock or well formation at a reasonable distance away from the wellbore. Skin is a dimensionless factor that quantifies any near-wellbore damage that might have occurred to the flow properties of the rock over the life of production of an artificially lifted well, known to contribute to the productivity of the well. Conventionally, this involves shutting in the well, i.e., ceasing production or flow from the well, and then monitoring the well using sensors such as pressure sensors. As is well known, when a well is closed after a long period of production, a slow increase in the shut-in wellbore pressure occurs, as the reservoir returns to equilibrium. Normally engineers use the change in this pressure over time to quantify reservoir characteristics. An example of this is what is known as a pressure buildup survey, also referred to as a build-up analysis. Analysis of the pressure and/or rate transient signals provides an indication of the productivity of the well and reflects the permeability of the reservoir formation, also referred to as formation permeability which is an indication of rock quality. Knowing the formation permeability surrounding a reservoir enables a reservoir operator or engineer to determine beam pump operating parameters that will result in enhanced productivity of the oilfield. Such operating parameters include amount of time the beam pump is running and amount of time the beam pump is off. Such on/off settings are controlled by cyclic pump off controllers. Knowing the formation permeability surrounding a reservoir also enables a reservoir operator or engineer to identify producing wells with poor productivity and potentially high skin, so that a work-over rig can be used to stimulate or remediate the well.
During reservoir production using intermittently produced beam pumping systems, pressure and rate transients occur because of the beam pump being turned on and off. These transients are much like the transient production that occurs when an artificially lifted well restarts after a period of inactivity. This transient production is known in the art as a drawdown test. Different types of drawdown tests can occur. In one case, referred to as constant rate production, the production rate is held constant and variation in bottomhole flowing pressure is observed. The “line source solution” as described in Lee, J., Rollins, J., and Spivey, J., [Pressure Transient Testing], SPE textbook series, Society of Petroleum Engineers (2003) to this flow condition for an infinite reservoir is one of the most widely used equations in pressure and rate transient analysis. When combined with superposition or convolution analysis, the solution can be applied to a wide range of producing wells. In a second type of drawdown test, referred to as constant pressure production, the pressure at the reservoir sandface is maintained at a constant while the production rate declines as a transient. This boundary condition is less well-known than the line source solution, however several solutions have been presented previously, as described in Ehlig-Economides, C. A., Well Test Analysis For Wells Produced At Constant Pressure, PhD thesis, Stanford University (1979). Unfortunately, these solutions are less mathematically convenient than the line source solution, and the inversion from Laplace space must be done numerically. Many simplifications have been proposed, and the simplest is that a rearrangement of the logarithmic line source solution can be acceptable at higher values of dimensionless time. See Earlougher, R. C., [Advances in Well Test Analysis], Society of Petroleum Engineers of AIME (1977).
As aging oilfields can have hundreds of beam pumped wells producing under artificial lift, it is not economical to perform a conventional well assessment test on all wells since this would result in lost production. It would be desirable to have a simpler, quicker, more cost-effective method for estimating the reservoir properties such as formation permeability and productivity of a producing well without the need for shutting in the well, particularly when pump off controllers are used.